Distribution and expected value of $X = \lfloor Y \rfloor$ where $Y \sim \exp(1)$.

Question

Find distribution and expected value of $X = \lfloor Y \rfloor$ where $Y \sim \exp(1)$.

In this case $\lfloor Y \rfloor$ is of course $\mathbb R_+ \cup \{0\} \rightarrow \mathbb Z_+\cup \{0\}$, so I get only integers as values.

Then:

$\mathbb P(X \leq t)= \mathbb P(\lfloor Y \rfloor \leq t) = \mathbb P(Y=i)$ for $i =0,1,2...$

so I guess it has something to do with a geometric distribution, am I right?

Answer 1

$P(\lfloor Y \rfloor =n)=P(n\leq Y <n+1)=e^{-n}-e^{-n-1}$. This gives the distrbution of the non-negative integer valued random variable $X$. $EX=\sum_{n=0}^{\infty} n(e^{-n}-e^{-n-1})$. will let you compute this sum.

The answer is $\frac 1 {e-1}$